Question

The area of bounded region R defined as \( R = \{(x,y): 0 < x < 2,\; 1 < y < 3,\; y > x \} \) is

• A. $\frac{3}{4}$
• B. $\frac{1}{2}$
• C. $\frac{7}{2}$
• D. $\frac{7}{8}$

Hint:

For regions bounded by simple lines, geometric subtraction (Total Area - Area of excluded shape) is often much faster and less error-prone than setting up double integrals.

Answer 1

The Correct Option is C

Area of the Region Defined by Inequalities

Step 1: Identify the region
The region \(R\) is defined by the inequalities:
1) \(0 < x < 2\) (Vertical strip between \(x=0\) and \(x=2\))
2) \(1 < y < 3\)   (Horizontal strip between \(y=1\) and \(y=3\))
3) \(y > x\)   (Region above the line \(y=x\))
The intersection of the first two gives a rectangle with vertices:
\((0,1), (2,1), (2,3), (0,3)\)

Step 2: Determine intersection with the line \(y=x\)
The line \(y=x\) intersects the rectangle at:
- At \(y=1\), \(x=1\)
- At \(y=3\), \(x=3\) (outside the strip \(x<2\))
- At \(x=0\), \(y=0\) (below \(y>1\))
- At \(x=2\), \(y=2\)
Hence, the triangle below the line inside the rectangle has vertices \((1,1), (2,1), (2,2)\).

Step 3: Compute areas
- Area of the rectangle \(= (2-0) \times (3-1) = 2 \times 2 = 4\)
- Area of the triangle \(= \frac{1}{2} \times base \times height = \frac{1}{2} \times (2-1) \times (2-1) = \frac{1}{2}\)

Step 4: Bounded area where \(y > x\)
\[ Area(R) = Area(Rectangle) - Area(Triangle) = 4 - \frac{1}{2} = \frac{7}{2} \]

Answer: The area of the region is \(\frac{7}{2}\), which is Option (3).